Question

Water rises to a height h in a capillary at the surface of earth. On the surface of the moon the height of water column in the capillary will be:-
A. 6h
B. $\dfrac{1}{6}$h
C. h
D. zero

Answer 1

Hint: We will use the formula for rise in height of a capillary tube. Compare the value of ‘h’ at the surface of earth and at the surface of the moon. Consider all other values to be constant except that of acceleration due to gravity ‘g’. Then find the value of ‘h’ at the surface of earth.

Formula used:
\[\text{h=}\dfrac{\text{2Tcos }\!\!\theta\!\!\text{ }}{\text{ }\!\!\rho\!\!\text{ rg}}\]

Complete step by step solution:
It has been scientifically proven that the acceleration due to gravity at the surface of the moon is $\dfrac{1}{6}\text{th}$ of the gravity at the surface of the earth, that is, ${{g}_{m}}=\dfrac{1}{6}{{g}_{e}}$. …………….(1)
Here, ${{g}_{m}}$ represents the gravity at the surface of the moon and ${{g}_{e}}$represents the gravity on the surface of the earth.
Now, we will use the formula of $\text{h=}\dfrac{\text{2Tcos }\!\!\theta\!\!\text{ }}{\text{ }\!\!\rho\!\!\text{ rg}}$
Where,
 h is the rise in height of liquid in the capillary tube ,
T is the surface tension of liquid, water in this case,
$\text{ }\!\!\theta\!\!\text{ }$ is the angle of contact,
$\text{ }\!\!\rho\!\!\text{ }$is the density of liquid (water),
 r is the radius of tube and,
 g is the acceleration due to gravity,
Now,
At the surface of the earth, ${{\text{h}}_{e}}\text{ }\!\!\alpha\!\!\text{ }\dfrac{\text{1}}{{{\text{g}}_{\text{e}}}}$ ………………(2)
At the surface of the moon, \[{{\text{h}}_{\text{m}}}\text{ }\!\!\alpha\!\!\text{ }\dfrac{\text{1}}{{{\text{g}}_{\text{m}}}}\] ………………(3)
Considering other values to be constant and dividing equation (2) by equation (3), we get
$\Rightarrow \dfrac{{{\text{h}}_{\text{e}}}}{{{\text{h}}_{\text{m}}}}\text{=}\dfrac{{{\text{g}}_{\text{m}}}}{{{\text{g}}_{\text{e}}}}$ …………..(4)
Now, we will use substitute equation (1) in equation (4), we get,
$\Rightarrow \dfrac{{{\text{h}}_{\text{e}}}}{{{\text{h}}_{\text{m}}}}\text{=}\dfrac{\text{1}}{\text{6}}$
$\therefore {{\text{h}}_{\text{m}}}\text{= 6}{{\text{h}}_{e}}$
Now, As per given in the question \[{{h}_{e}}=h\].
$\Rightarrow {{\text{h}}_{\text{m}}}\text{=6h}$

So, the correct answer is “Option A”.

Note: Remember the relation between acceleration due to gravity at surface of earth and moon that is \[{{\text{g}}_{\text{m}}}\text{=}\dfrac{\text{1}}{\text{6}}\text{g}{}_{\text{h}}\].There can be question where we are asked to find the rise of height in capillary tube on some other planet. Then the relation between acceleration due to gravity on that planet and earth will change and we must consider it accordingly.